What is the official definition of $S^1$

Question

I need an official definition of $S^1$ that is better than $\{circle\}$.

The reason is because I am interesting in defining a function $f: \mathbb{R} \to S^1$ where $\mathbb{R}$ is the interval $[0, 2\pi)$, but I do not know what the image is

I tried to come up with one:

$S^1 = \{x|\|x\| = 1\}$

But there seems to be better ones out there.

Can someone please provide with an official def of $S^1$?

Answer 1

In general, $S^{n}=\{ \boldsymbol{x} \in \mathbb{R}^{n+1}: |\boldsymbol{x}| =1 \}$.

$S^{0}$ is two isolated points, $S^{1}$ is a circle, $S^{2}$ is a sphere and $S^{3}$ is a 3-sphere.

Moreover, a torus $T^{2}$ is $S^{1}\times S^{1}$ and (infinite) cylinder is $\mathbb{R}^{1}\times S^{1}$.

See also the topological definition here.

Answer 2

There are several definitions, which are all in some sense equivalent. People use the definition that makes thing easier to write down and that depends on what they want to do.

If you work in complex analysis, you might find it nice to set $S^1=\{e^{i2\pi x} : x\in [0,1)\}\subset\mathbb{C}$.

If you do geometry you might go with the definition you presented, with ambient space $\mathbb{R}^2$.

If you work in topology you would probably choose a definition that does not make use of an ambient space or a norm.

There are also a lot of other fields that like the complex analysis definition, as it turns the circle into an object with an algebraic structure: multiplication (it becomes a group).

All these definitions are equivalent in SOME sense. For example, they all have (canonical) topologies, which are homoeomorphic (i.e., equivalent in the sense of topology). As another example, the first two have (canonical) geometric structures, which are isomorphic (equivalent in the sense of geometry).